Combinatorics and algorithms of geometric arrangements

Abstract

In this talk we survey the most important combinatorial and algorithmic results on arrangements. Arrangements are the subdivisions of Euclidean space defined by collections of algebraic manifolds. They have turned out to be a crucial concept in Computational Geometry with numerous unforeseen applications to a wide variety of fundamental geometric problems. We will discuss some key areas, such as zone and lower envelope theorems, many cell problems, and point location. Many of the fundamental algorithmic techniques in geometry can be illustrated on problems in arrangements. Examples include randomized algorithms and epsilon-nets, parametric search, partition trees, and duality.